Prove each statement in 6-9 using mathematical induction.  9. For every integer n >=3, 43 + 44 + 45 + … + 4n =4(4n-16)/3

Use mathmatical induction to prove the following statement true for every integer n. 2+6+18+...+ 2 * 3n-1 = 3n - 1

Use the principle of mathematical induction to show that the statement is true for all natural numbers. 2² +4² +6² + ... + (2n)² = 2n(n + 1)(2n + 1) 3 Let Pn denote the statement: 22 +4² +6² + ... + (2n)² + Check that P₁ is true: 2² = 4 and 2(C Assume Pk is true: 22 +42 +62 + + (² 2 (2n)² = 2n(n + 1)(2n + 1) 3 2² +4² +6² + ... + (2(k+1))² ))( + ¹)(² 1) (2 1 3 To show that Pk+1 is true, add (2(k + 1))² to both sides of Pk. 2² + 4² + 6² + ... + (2k)² + (2( ))²³ = 2 2 |)²-² = +1 1) 2k(k + 1)(2k + 1) 12( + 3 = Rewrite the right-hand side as a single fraction, and then factor the numerator completely. 3 3 2k(k + 1)(2k + 1) +(2( 3 (k + 1)(2k + 1) Thus P₁ is true. 2 2 :))²

Prove the given statement by using mathematical induction. n n 1 + 1 + 1 + - + (-) - ¹ - (-)" = 1 2 4 8